= Conjugacy classes of the symmetric group on three letters
{title2=$S_3:\ 1+3+2=6$}
The <symmetric group> $S_3$ has exactly six <permutations>, written using <permutation cycles>:
$$
S_3=\{e,(12),(13),(23),(123),(132)\}.
$$
Here $e$ is the <identity element>; $(12)$, $(13)$ and $(23)$ are <transpositions>; and $(123)$ and $(132)$ are <permutation cycles> of length three. With $t=(12)$ and $r=(123)$, rightmost-first <function composition> gives $r^2=(132)$ and $r^{-1}=r^2$. The <conjugacy classes> and representative <centralizers> are
$$
\begin{array}{c|c|c|c}
x&\operatorname{Cl}_{S_3}(x)&C_{S_3}(x)&|\operatorname{Cl}_{S_3}(x)|\\
\hline
e&\{e\}&S_3&6/6=1\\
t&\{(12),(13),(23)\}&\{e,t\}&6/2=3\\
r&\{(123),(132)\}&\{e,r,r^2\}&6/3=2
\end{array}
$$
\Image[/s3-conjugacy-classes.png]
{title=All six <permutations> of the <symmetric group> $S_3$, their <conjugacy classes>, and representative <centralizers>}
An explicit check of the <conjugation> $gtg^{-1}$ with $g=r=(123)$ sends $1\mapsto3\mapsto3\mapsto1$, $2\mapsto1\mapsto2\mapsto3$, and $3\mapsto2\mapsto1\mapsto2$, where each chain applies $g^{-1}$, then $t$, then $g$. Thus $1$ is fixed and $2,3$ are exchanged: $gtg^{-1}=(23)$. Relabelling also produces $(13)$, so the three <transpositions> share one <conjugacy class>. A <transposition> reverses the length-three <permutation cycle>, since $trt^{-1}=(213)=(132)=r^{-1}$, putting both length-three <permutation cycles> in one <conjugacy class>.
Checking each of the six possible <permutations> against $gx=xg$ gives the displayed <centralizers>. Only $e$ has a one-element <conjugacy class>, so the <center of a group> here is $Z(S_3)=\{e\}$. The <class equation> is $6=1+3+2$. The second diagram in <centralizer and normalizer> uses the same $r$ and $t$ to show how preserving a <cyclic subgroup> differs from fixing its <group generator>.
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